We are working with the Hilbert space $L^2(\mu)$, such that $\mu(X)<\infty$.
If $M$ is the subspace of constant functions, define $M^\perp$ both formally and "intuitively". (This is a homework problem). My problem is the intuitive description part.
We are told that
$$M := \{g \in L^2(\mu) : g(x) = c \in \mathbb{R}, \ \forall x \in X\}$$ Therefore, if we want the orthogonal complement,
$$M^\perp= \{f \in L^2(\mu) : \langle f,g\rangle=0, g \in M\}$$
Therefore, for any function $f \in M^\perp$, and real constant $c$
$$\langle f, c\rangle = 0 = \int_X cf \ \mathrm{d} \mu = c \int_X f \ \mathrm{d} \mu$$
Thus, it follows that
$$M^\perp = \left\{f \in L^2(\mu) : \int_X f \ \mathrm{d} \mu = 0 \right\}$$
But, I don't understand what would an intuitive definition/description would be in this case.